It is reasonable to say that a definitive solution of Asser's conjecture seems still far away.
Thus, it is natural to consider the spectra of some restricted classes of first-order logic.
Fagin [1]} was the first to notice that to settle Asser's conjecture,
it is sufficient to consider only first-order logic over graphs.
More formally, he showed that
for every spectrum \(A\) , there is a positive integer \(k>0\)  such that
\(\lbrace n^k \mid n \in A\rbrace \)  is the spectrum of a sentence using only one binary relation symbol.
Implicitly, it implies that if there is a spectrum whose complement is not a spectrum,
then there is such a spectrum of first-order sentence using only one binary relation [2]}, [3]},
i.e., Asser's conjecture can be reduced to first-order sentences over graphs.
